Multiple arbitrarily inflated count regression models

This thesis addresses the common challenge of modeling count data that exhibits inflation at multiple and arbitrary values, a feature often inadequately captured by standard statistical models. To flexibly model these complex data structures, this work introduces a unified family of models and their corresponding regression extensions that advance the existing literature. First, an extended Poisson distribution is developed, which encompasses numerous nested inflated Poisson models as special cases. This framework is then generalized to the Negative Binomial family to account for overdispersion. Finally, as an alternative to these two approaches, a novel discrete model based on a modified Uniform-Geometric distribution is presented. For each proposed model family, fundamental distributional properties are derived, maximum likelihood estimation methods based on Fisher scoring and the EM algorithm are developed, and an inferential framework is established. Simulation studies and real-data applications demonstrate the flexibility, superior goodness-of-fit, and enhanced interpretability of the proposed models compared to existing methods. This thesis provides a comprehensive and applicable methodological toolkit for modeling arbitrary inflation patterns in count data, making a significant contribution to the field.